Inverted-pendulum Model for the Stability of Bouncing Droplets on Vibrating Foundation

We investigate experimentally liquid (oil) droplets vibrating over a bath of the same liquid and relate the phenomenon to the well-known effect of the stabilization of an inverted pendulum on a vibrating foundation (a “Kapitza pendulum”). Small fast vibrations can be substituted by an effective “levitation” force. We further discuss this effective stabilizing “spring force”. Besides the inverted pendulum, such a force is responsible for the stability of an “Indian rope” and for rising of “cornstarch monsters.” We suggest a simple model relating the effect to the parametric resonance as described by the Mathieu equation, which explains stabilization of an inverted pendulum with vibration foundation. We discuss how the mathematical separation of the slow and fast motion provides insights on the relation of vibro-levitation of oil droplets and soft materials with the vibro-stabilization of an inverted pendulum, and the “Indian rope” and “Cornstarch monster” tricks. The effect can be used in the design of novel materials.